Finite Group Databasebeta

← #10596182

Subgroups — #10596182

Order 576.

756 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 171 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#126C211yesmaximal, characteristic, complemented
2#64104C213yesmaximal, complemented
3#64103C213yesmaximal, complemented
4#64093C213yesmaximal, complemented
5#64091C213yesmaximal, complemented
6#64085C211yesmaximal, characteristic, complemented
7#64083C211yesmaximal, characteristic, complemented
8#5711—34nomaximal
9C23×C3⋊D3C2211yescharacteristic
10C2×#638C2213yes—
11C2×C4⋊(C3⋊D3)C2213yes—
12C2×C4⋊(C3⋊D3)C2213yes—
13C2×C4⋊(C3⋊D3)C2213yescomplemented
14C2×C4×C3⋊D3C2213yescomplemented
15C32×C4⋊C4C2213yescomplemented
16C32×C22⋊C4C2213yes—
17C32×C42C2211yescharacteristic, complemented
18#3752C2213yes—
19#3752C2213yes—
20#3752C2213yescomplemented
21#3750C2213yescomplemented
22C3×#126D314yescomplemented
23#895—312no—
24#894—312no—
25#884—312no—
26#882—312no—
27#876—34no—
28#874—34no—
29C23×C32C2311yescharacteristic
30C22×C3⋊D3—43no—
31C22×C3⋊D3C2311yescharacteristic
32C22×C3⋊D3C2311yescharacteristic
33C22×C3⋊D3C2313yes—
34C2×C32×C4C2313yes—
35C2×C32×C4C2313yes—
36#638—43no—
37C2×C3⋊Dic3C2313yes—
38C4⋊(C3⋊D3)—43no—
39C4⋊(C3⋊D3)—43no—
40C4⋊(C3⋊D3)—43no—
41C4×C3⋊D3—43no—
42#560—91noSylow
43C23×D3—34no—
44C2×#67—312no—
45C2×D12—312no—
46C2×D12—312no—
47C2×D12—312no—
48C2×C4×D3—312no—
49C3×C4⋊C4C2×D3112yescomplemented
50C3×C22⋊C4C2×D3112yes—
51C3×C42C2×D314yescomplemented
52(C2×D3)⋊C4—312no—
53(C2×D3)⋊C4—312no—
54(C2×D3)⋊C4—312no—
55Dic3⋊C4—312no—
56C22×C32—23no—
57C22×C32C2411yescharacteristic
58C2×C3⋊D3—23no—
59C2×C3⋊D3—23no—
60C2×C3⋊D3—23no—
61C2×C3⋊D3—23no—
62C2×C3⋊D3—23no—
63C2×C3⋊D3—44no—
64C32×C4—23no—
65C32×C4—23no—
66C3⋊Dic3—23no—
67#127—91no—
68#126—91no—
69#126C3⋊D311yescharacteristic, complemented
70#123—93no—
71#121—93no—
72#121—93no—
73#120—93no—
74C23×C3C22×D314yes—
75C22×D3—34no—
76C22×D3—34no—
77C22×D3—312no—
78C22×D3—1212no—
79C2×C3×C4C22×D3112yes—
80C2×C3×C4C22×D3112yes—
81#67—1212no—
82C2×Dic3—312no—
83D12—1212no—
84D12—1212no—
85D12—1212no—
86C4×D3—1212no—
87C2×C32—41no—
88C2×C32H(5,𝔽2)13yes—
89C3⋊D3—43no—
90C3⋊D3—41no—
91C3⋊D3—41no—
92C24—91no—
93C2×D4—93no—
94C2×D4—93no—
95C2×D4—93no—
96C2×D4—93no—
97C22×C4—93no—
98C4⋊C4—93no—
99C4⋊C4C2×C3⋊D313yescomplemented
100C22⋊C4—93no—
101C22⋊C4—93no—
102C22⋊C4—93no—
103C22⋊C4C2×C3⋊D313yes—
104C42C2×C3⋊D311yescharacteristic, complemented
105C22×C3—212no—
106C22×C3C23×D314yes—
107C2×D3—612no—
108C2×D3—612no—
109C2×D3—612no—
110C2×D3—612no—
111C2×D3—612no—
112C2×D3—1216no—
113C3×C4—212no—
114C3×C4—212no—
115Dic3—612no—
116C32#56011yescharacteristic, complemented, Sylow
117C23—91no—
118C23—91no—
119C23—93no—
120C23—363no—
121C23C22×C3⋊D311yescharacteristic
122D4—363no—
123D4—363no—
124D4—363no—
125D4—363no—
126C2×C4—93no—
127C2×C4—363no—
128C2×C4C22×C3⋊D313yes—
129C2×C4C22×C3⋊D313yes—
130C2×C3—44no—
131C2×C3#1009112yes—
132D3—124no—
133D3—124no—
134D3—1212no—
135C22—23no—
136C22—183no—
137C22—183no—
138C22—183no—
139C22—183no—
140C22—183no—
141C22—364no—
142C22C23×C3⋊D311yescharacteristic
143C4—23no—
144C4—23no—
145C4—183no—
146C3#571114yescomplemented
147C2—41no—
148C2—361no—
149C2—361no—
150C2—363no—
151C2#6436613yes—
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×#126C2
hypercenter#126C3⋊D3
hypocenterC32#560
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #126

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C32×#126C2semidirect
2. #64104C2semidirect
3. #64103C2semidirect
4. #64093C2semidirect
5. #64091C2semidirect
6. #64085C2semidirect
7. #64083C2semidirect
13. C2×C4⋊(C3⋊D3)C22semidirect
14. C2×C4×C3⋊D3C22semidirect
15. C32×C4⋊C4C22semidirect
17. C32×C42C22semidirect
20. #3752C22semidirect
21. #3750C22semidirect
22. C3×#126D3semidirect
49. C3×C4⋊C4C2×D3semidirect
51. C3×C42C2×D3semidirect
69. #126C3⋊D3semidirect
99. C4⋊C4C2×C3⋊D3semidirect
104. C42C2×C3⋊D3semidirect
116. C32#560semidirect
146. C3#5711semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C32×#1262
#641042
#641032
#640932
#640912
#640852
#640832
#57113